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

theorem Th22:
  for x,y being Element of [:REAL,REAL:] holds Eukl_dist2.(x,y) = 0 iff x = y
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];
  thus Eukl_dist2.(x,y) = 0 implies x = y
  proof
    set d2 = real_dist.(x2,y2);
    set d1 = real_dist.(x1,y1);
    assume Eukl_dist2.(x,y) = 0;
    then
A2: sqrt(d1^2 + d2^2) = 0 by A1,Def18;
A3: 0 <= d1^2 & 0 <= d2^2 by XREAL_1:63;
    then d1 = 0 by A2,Lm1;
    then
A4: x1 = y1 by METRIC_1:8;
    d2 = 0 by A2,A3,Lm1;
    hence thesis by A1,A4,METRIC_1:8;
  end;
  assume x = y;
  then
A5: (real_dist.(x1,y1))^2 = 0^2 & (real_dist.(x2,y2))^2 = 0^2 by METRIC_1:8;
  Eukl_dist2.(x,y) = sqrt((real_dist.(x1,y1))^2 + (real_dist.(x2,y2) ) ^2
  ) by A1,Def18
    .= 0 by A5;
  hence thesis;
end;
