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

theorem Th28:
  for x,y being Element of [:REAL,REAL,REAL:] holds
  Eukl_dist3.(x,y) = 0 iff x = y
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];
  thus Eukl_dist3.(x,y) = 0 implies x = y
  proof
    set d3 = real_dist.(x3,y3);
    set d2 = real_dist.(x2,y2);
    set d1 = real_dist.(x1,y1);
A2: 0 <= d2^2 & 0 <= d3^2 by XREAL_1:63;
    assume Eukl_dist3.(x,y) = 0;
    then sqrt(d1^2 + d2^2 + d3^2) = 0 by A1,Def22;
    then
A3: sqrt(d1^2 + (d2^2 + d3^2)) =0;
    0 <= d2^2 & 0 <= d3^2 by XREAL_1:63;
    then
A4: 0 <= d1^2 & 0 + 0 <= d2^2 + d3^2 by XREAL_1:7,63;
    then d1 = 0 by A3,Lm1;
    then
A5: x1 = y1 by METRIC_1:8;
A6: d2^2 + d3^2 = 0 by A3,A4,Lm1;
    then d2 = 0 by A2,XREAL_1:27;
    then
A7: x2 = y2 by METRIC_1:8;
    d3 = 0 by A6,A2,XREAL_1:27;
    hence thesis by A1,A5,A7,METRIC_1:8;
  end;
  assume
A8: x = y;
  then
A9: (real_dist.(x1,y1))^2 = 0^2 & (real_dist.(x2,y2))^2 = 0^2 by METRIC_1:8;
  Eukl_dist3.(x,y) = sqrt((real_dist.(x1,y1))^2 + (real_dist.(x2,y2))^2 +
  (real_dist.(x3,y3))^2) by A1,Def22
    .= 0^2 by A8,A9,METRIC_1:8,SQUARE_1:17;
  hence thesis;
end;
