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

theorem Th12: :::
  for a,b,c,d being Real st 0 <= a & 0 <= b & 0 <= c & 0 <= d holds
    sqrt((a + c)^2 + (b + d)^2) <= sqrt(a^2 + b^2) + sqrt(c^2 + d^2)
proof
  let a,b,c,d be Real;
  assume 0 <= a & 0 <= b & 0 <= c & 0 <= d;
  then 0 <= a*c & 0 <= d*b by XREAL_1:127;
  then
A1: 0 + 0 <= a*c + d*b by XREAL_1:7;
  0 <= d^2 & 0 <= c^2 by XREAL_1:63;
  then
A2: 0 + 0 <= c^2 + d^2 by XREAL_1:7;
  then
A3: 0 <= sqrt(c^2 + d^2) by SQUARE_1:def 2;
  0 <= ((a*d) - (c*b))^2 by XREAL_1:63;
  then 0 <= (a^2*d^2 + c^2*b^2) - 2*(a*d)*(c*b);
  then 0 + 2*(a*d)*(c*b) <= (a^2*d^2 + c^2*b^2) by XREAL_1:19;
  then (b^2*d^2) + 2*(a*d)*(c*b) <= (a^2*d^2 + c^2*b^2) + (b^2*d^2 ) by
XREAL_1:6;
  then
A4: (a^2*c^2) + ((b^2*d^2) + (2*(a*d)*(c*b))) <= (a^2*c^2) + ((b^2*d^2) + (a
  ^2*d^2 + c^2*b^2)) by XREAL_1:6;
  0 <= a^2 & 0 <= b^2 by XREAL_1:63;
  then
A5: 0 + 0 <= a^2 + b^2 by XREAL_1:7;
  then 0 <= sqrt(a^2 + b^2) by SQUARE_1:def 2;
  then
A6: 0 + 0 <= sqrt(a^2 + b^2) + sqrt(c^2 + d^2) by A3,XREAL_1:7;
  0 <= (a*c + d*b)^2 by XREAL_1:63;
  then sqrt((a*c + d*b)^2) <= sqrt((a^2 + b^2)*(c^2 + d^2)) by A4,SQUARE_1:26;
  then 2*sqrt((a*c + d*b)^2) <= 2*(sqrt((a^2 + b^2)*(c^2 + d^2))) by XREAL_1:64
;
  then 2*sqrt((a*c + d*b)^2) <= 2*(sqrt(a^2 + b^2)*sqrt(c^2 + d^2)) by A5,A2,
SQUARE_1:29;
  then 2*(a*c + d*b) <= 2*(sqrt(a^2 + b^2)*sqrt(d^2 + c^2)) by A1,SQUARE_1:22;
  then b^2 + 2*(a*c + d*b) <= 2*(sqrt(a^2 + b^2)*sqrt(d^2 + c^2)) + b^2 by
XREAL_1:6;
  then
  d^2 + (b^2 + 2*(a*c + d*b)) <= d^2 + (b^2 + 2*(sqrt(a^2 + b^2)*sqrt(d^2
  + c^2))) by XREAL_1:6;
  then
  c^2 + (d^2 + (b^2 + 2*(a*c + d*b))) <= (d^2 + (b^2 + 2*(sqrt(a^2 + b^2)
  *sqrt(d^2 + c^2)))) + c^2 by XREAL_1:6;
  then a^2 + (c^2 + (d^2 + ((b^2 + 2*(d*b)) + 2*(a*c)))) <= a^2 + (c^2 + (d^2
  + (b^2 + 2*(sqrt(a^2 + b^2)*sqrt(d^2 + c^2))))) by XREAL_1:6;
  then (a + c)^2 + (d + b)^2 <= (a^2 + b^2) + 2*(sqrt(a^2 + b^2)*sqrt(c^2 + d
  ^2)) + (c^2 + d^2);
  then
  (a + c)^2 + (d + b)^2 <= (sqrt(a^2 + b^2))^2 + 2*(sqrt(a^2 + b^2)*sqrt(
  c^2 + d^2)) + (c^2 + d^2) by A5,SQUARE_1:def 2;
  then
A7: (a + c)^2 + (d + b)^2 <= (sqrt(a^2 + b^2))^2 + 2*sqrt(a^2 + b^2)*sqrt(c
  ^2 + d^2) + (sqrt(c^2 + d^2))^2 by A2,SQUARE_1:def 2;
  0 <= (a + c)^2 & 0 <= (d + b)^2 by XREAL_1:63;
  then 0 + 0 <= (a + c)^2 + (d + b)^2 by XREAL_1:7;
  then
  sqrt((a + c)^2 + (d + b)^2) <= sqrt((sqrt(a^2 + b^2) + sqrt(c^2 + d^2))
  ^2) by A7,SQUARE_1:26;
  hence thesis by A6,SQUARE_1:22;
end;
