 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem Th1:
  for X be RealUnitarySpace,
      x,y be Point of X,
      z,t be Point of MetricSpaceNorm RUSp2RNSp X
    st x=z & y = t holds
   ||.x-y.|| = dist(z,t)
proof
  let X be RealUnitarySpace,
      x,y be Point of X,
      z,t be Point of MetricSpaceNorm RUSp2RNSp X;
  assume A1: x=z & y = t;
  reconsider x1=x,y1=y as Point of RUSp2RNSp X;
  thus ||.x-y.|| = ||.x1-y1.|| by DUALSP04:3,DUALSP04:2
                .= dist(z,t) by A1,NORMSP_2:def 1;
end;
