reserve a,b,x,y for Real,
  i,k,n,m for Nat;
reserve p,w for Real;

theorem Th1:
  for A being SubSpace of RealSpace, p,q being Point of A st x = p
  & y = q holds dist(p,q) = |.x-y.|
proof
  let A be SubSpace of RealSpace, p,q be Point of A;
  assume
A1: x = p & y = q;
  thus dist(p,q) = (the distance of A).(p,q) by METRIC_1:def 1
    .= (real_dist).(x, y) by A1,METRIC_1:def 13,TOPMETR:def 1
    .= |.x-y.| by METRIC_1:def 12;
end;
