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

theorem Th29:
  for x,y being Element of [:REAL,REAL,REAL:] holds Eukl_dist3.(x,
  y) = Eukl_dist3.(y,x)
proof
  let x,y be Element of [:REAL,REAL,REAL:];
  reconsider x1 = x`1_3, x2 = x`2_3, x3 = x`3_3,
      y1 = y`1_3, y2 = y`2_3, y3 = y`3_3 as
  Element of REAL;
A1: x = [x1,x2,x3] & y = [y1,y2,y3];
  then
  Eukl_dist3.(x,y) = sqrt((real_dist.(x1,y1))^2 + (real_dist.(x2,y2) ^2) +
  (real_dist.(x3,y3)^2)) by Def22
    .= sqrt((real_dist.(y1,x1))^2 + (real_dist.(x2,y2)^2) + (real_dist.(x3,
  y3)^2)) by METRIC_1:9
    .= sqrt((real_dist.(y1,x1))^2 + (real_dist.(y2,x2)^2) + (real_dist.(x3,
  y3)^2)) by METRIC_1:9
    .= sqrt((real_dist.(y1,x1))^2 + (real_dist.(y2,x2)^2) + (real_dist.(y3,
  x3)^2)) by METRIC_1:9
    .= Eukl_dist3.(y,x) by A1,Def22;
  hence thesis;
end;
