reserve u,v,x,y,z,X,Y for set;
reserve r,s for Real;

theorem Th1:
  for a,b,c,d being Real holds
  (a+b)^2+(c+d)^2 <= (sqrt(a^2+c^2)+sqrt(b^2+d^2))^2
  proof
    let a,b,c,d be Real;
    set A = a^2+c^2, B = b^2+d^2, C1 = sqrt(A), C2 = sqrt(B);
A1: 0 <= C1 & 0 <= C2 by SQUARE_1:def 2;
A2: C1^2 = A & C2^2 = B by SQUARE_1:def 2;
A3: a^2+2*a*b+b^2 + (c^2+2*c*d+d^2) = a^2+c^2+b^2+d^2+(2*a*b+2*c*d);
A4: A+2*C1*C2+B = A+B+2*C1*C2;
A5: 2*a*b+2*c*d = 2*(a*b+c*d);
A6: 2*(C1*C2) = 2*C1*C2;
    (a*d-c*b)^2 = (a*d)^2+(c*b)^2-2*(a*d)*(c*b);
    then
A7: 0+2*(a*d)*(c*b) <=
    (a*d)^2+(c*b)^2-2*(a*d)*(c*b)+2*(a*d)*(c*b) by XREAL_1:6;
A8: A*B = (sqrt(A*B))^2 by SQUARE_1:def 2;
    a^2*b^2+c^2*d^2+(2*a*b*c*d) <= a^2*b^2+c^2*d^2+(c^2*b^2+a^2*d^2)
    by A7,XREAL_1:6;
    then
A9: (a*b+c*d)^2 <= A*B;
    C1*C2 = sqrt(A*B) by SQUARE_1:29;
    then a*b+c*d <= C1*C2 by A1,A8,A9,SQUARE_1:16;
    then 2*a*b+2*c*d <= 2*C1*C2 by A5,A6,XREAL_1:64;
    hence thesis by A2,A3,A4,XREAL_1:6;
  end;
