
theorem THYQ:
  for a,b,c,d being Real holds dist(|[a,b]|,|[c,d]|) = sqrt ((a-c)^2+(b-d)^2)
  proof
    let a,b,c,d be Real;
A1: |[a,b]|`1 = a & |[a,b]|`2 = b & |[c,d]|`1 = c & |[c,d]|`2 = d by EUCLID:52;
    reconsider P = |[a,b]|,Q = |[c,d]| as Point of Euclid 2
      by EUCLID:22;
    dist(|[a,b]|,|[c,d]|) = dist(P,Q) by TOPREAL6:def 1
       .= (Pitag_dist 2).(P,Q) by METRIC_1:def 1
       .= sqrt ((a-c)^2+(b - d)^2) by A1,TOPREAL3:7;
    hence thesis;
  end;
