reserve X, Y, Z, W for non empty MetrSpace;
reserve X,Y for non empty MetrSpace;
reserve Z for non empty MetrSpace;

theorem Th23:
  for x,y being Element of [:REAL,REAL:] holds Eukl_dist2.(x,y) =
  Eukl_dist2.(y,x)
proof
  let x,y be Element of [:REAL,REAL:];
  reconsider x1 = x`1, x2 = x`2, y1 = y`1, y2 = y`2 as Element of REAL;
A1: x = [x1,x2] & y = [y1,y2];
  then
  Eukl_dist2.(x,y) = sqrt((real_dist.(x1,y1))^2 + (real_dist.(x2,y2) ^2))
  by Def18
    .= sqrt((real_dist.(y1,x1))^2 + (real_dist.(x2,y2)^2)) by METRIC_1:9
    .= sqrt((real_dist.(y1,x1))^2 + (real_dist.(y2,x2)^2)) by METRIC_1:9
    .= Eukl_dist2.(y,x) by A1,Def18;
  hence thesis;
end;
